Proper way to clean the air handler drain lines

Hi all, I hope someone here can offer some guidance for how to clean an air handler. Living in South Florida, the AC is usually taxed heavily 9 months out of the year. I noticed earlier this year that gooey gunk started building up in the air handler drainage channel, causing condensation/water to "overflow" onto the floor.
To this point, I have been using a couple of small electrical wires as plumbing snake to clear channels - just to keep the gunk from building up. But the wire doesn't reach all the way to the back and I'm thinking there is got to be a more thorough way to cleaning those drains. And as you can see from the pix, there really isn't much room to work with. The slot to access the underside of the handler is basically the slot for the air filter. Can anyone help? Many thanks!